When a company are searching for new F/Os, and they are searching for those with a TR and several hours on the type, Is there a chance for me if I apply for the job without the TR and/or the hours on the type or am I just going to irritate the one taking care of the resumes?
Depends on who you apply to. Some might argue that having a TR makes you more employable but it is also a huge financial risk with no guarantee of a job. I wouldn't rule out applying just because you dont have a type rating unless the job requirements specifically say that you need one. I personally became a flight instructor and did my IR after I had over 1000hrs total time, then applied for airline jobs. I was flying a small turboprops within 4 months.
I also wonder how you have experienced companies attitude against woman pilots. Have you seen any quotation? Or is it harder for a woman?
Should make no difference at all. Plenty of female pilots where I work flying 737s.
Best of luck!